Speed-Up Techniques for Shortest-Path Computations - CiteSeerX

Speed-Up Techniques for Shortest-Path Computations? Dorothea Wagner and Thomas Willhalm Universit¨ at Karlsruhe (TH) Fakult¨ at f¨ ur Informatik Institut f¨ ur Theoretische Informatik D-76128 Karlsruhe {wagner,willhalm}@ira.uka.de http://i11www.informatik.uni-karlsruhe.de/

Abstract. During the last years, several speed-up techniques for Dijkstra’s algorithm have been published that maintain the correctness of the algorithm but reduce its running time for typical instances. They are usually based on a preprocessing that annotates the graph with additional information which can be used to prune or guide the search. Timetable information in public transport is a traditional application domain for such techniques. In this paper, we provide a condensed overview of new developments and extensions of classic results. Furthermore, we discuss how combinations of speed-up techniques can be realized to take advantage from different strategies.

1

Introduction

Computing shortest paths is a base operation for many problems in traffic applications. The most prominent are certainly route planning systems for cars, bikes and hikers, or timetable information systems for scheduled vehicles like trains and busses. If such a system is realized as a central server, it has to answer a huge number of customer queries asking for their best itineraries. Users of such a system continuously enter their requests for finding their “best” connections. Furthermore, similar queries appear as sub-problems in line planning, timetable generation, tour planning, logistics, and traffic simulations. The algorithmic core problem that underlies the above scenario is a special case of the single-source shortest-path problem on a given directed graph with non-negative edge lengths. While this is obvious for route planning in street networks, different models and approaches have been presented to solve timetable information by finding shortest paths in an appropriately defined graph. The typical problem to be solved in timetable information is “given a departure and an arrival station as well as a departure time, which is the connection that arrives as early as possible at the arrival station?”. There are two main approaches for ?

Partially supported by the Future and Emerging Technologies Unit of EC (IST priority 6th FP) under contract no. FP6-021235-2 (project ARRIVAL).

modeling timetable information as shortest path problem, the time-expanded and the time-dependent approach. For an overview of models and algorithms for optimally solving timetable information we refer to [28]. In any case the particular graphs considered are huge, especially if the model used for timetable information expands time by modelling each event by a single vertex in the graph. Moreover, the number of queries to be processed within very short time is huge as well. This motivates the use of speed-up techniques for shortest-path computations. The main focus is to reduce the response time for on-line queries. In this sense, a speed-up technique is considered as a technique to reduce the search space of Dijkstra’s algorithm e.g. by using precomputed information or inherent information contained in the data. Actually, often the underlying data contain geographic information, that is a layout of the graph is provided. Furthermore, in many applications the graph can be assumed to be static, which allows a preprocessing. Due to the size of the graphs considered in route planning or timetable information and the fact that those graphs are typically sparse, preprocessing space requirements are only acceptable to be linear in the number of nodes. In this paper, we provide a systematic classification of common speed-up techniques and combinations of those. Our main intention is to give a concise overview of the current state of research. We restrict our attention to speed-up techniques where the correctness of the algorithms is guaranteed, i.e., that provably return a shortest path. However, most of them are heuristic with respect to the running time. More precisely, in the worst case, the algorithm with speed-up technique can be slower than the algorithm without speed-up technique. But experimental studies showed–sometimes impressive–improvements concerning the search front and consequently the running time. For most of these techniques, experimental results for different real-world graphs as well as generated graphs have been reported. However, as the effectiveness of certain speed-up techniques strongly depends on the graph data considered, we do not give a comparison of the speed-ups obtained. But we want to refer to the 9th DIMACS Implementation Challenge - Shortest Paths where also experiments on common data sets were presented [7]. In the next section, we will provide some formal definitions and a description of Dijkstra’s algorithm. Section 3 presents a classification of speed-up techniques for Dijkstra’s algorithm and discusses how they can be combined.

2 2.1

Preliminaries Definitions

A (directed) graph G is a pair (V, E), where V is a finite set of nodes and E is a set of edges, where an edge is an ordered pair (u, v) of nodes u, v ∈ V . Throughout this paper, the number of nodes |V | is denoted by n and the number of edges |E| is denoted by m. For a node u ∈ V , the number of outgoing edges |{(u, v) ∈ E}| is called the degree of the node. A path in G is a sequence of nodes (u1 , . . . , uk )

1 for all nodes u ∈ V set dist(u) := ∞ 2 initialize priority queue Q with source s and set dist(s) := 0 3 while priority queue Q is not empty 4 get node u with smallest tentative distance dist(u) in Q 5 for all neighbor nodes v of u 7 set new-dist := dist(u) + w(u, v) 8 if new-dist < dist(v) 9 if dist(v) = ∞ 10 insert neighbor node v in Q with priority new-dist 11 else 12 set priority of neighbor node v in Q to new-dist 13 set dist(v) := new-dist

Algorithm 1: Dijkstra’s algorithm. such that (ui , ui+1 ) ∈ E for all 1 ≤ i < k. A path with u1 = uk is called a cycle. Given edge weights l : E → R (“lengths”), theP length of a path P = (u1 , . . . , uk ) is the sum of the lengths of its edges l(P ) := 1≤i 0. [37] presents how graph-drawing algorithms help in the case where a layout of the graph is not given beforehand. This approach can be extended in a straight forward manner to other metric spaces than (R2 , k·k). In particular, it is possible to use more than two dimensions or other metrics like the Manhattan metric. Finally, the expensive square root function to compute the Euclidean distance can be replaced by an approximation.

Landmarks With preprocessing, it is possible to gather information about the graph that can be used to obtain improved lower bounds. In [10], a small fixedsized subset L ⊂ V of “landmarks” is chosen. Then, for all nodes v ∈ V , the distance d(v, l) to all nodes l ∈ L is precomputed and stored. These distances can be used to determine a feasible potential. For each landmark l ∈ L, we (l) define the potential pt (v) := d(v, l) − d(t, l). Due to the triangle inequality (l) d(v, l) ≤ d(v, t) + d(t, v), the potential pt is feasible and indeed a lower bound for the distance to t. The potential is then defined as the maximum over all (l) potentials: pt (v) := max{pt (v); l ∈ L}. It is easy to show that the maximum of feasible potentials is again a feasible potential. For landmarks that are situated next to or “behind” the target t, the lower (l) bound pt (u) should be fairly tight, as shortest paths to t and l most probably share a common sub-path. Landmarks in other regions of the graph however, may attract the search to themselves. This insight justifies to consider, in a specific (l) search from s to t, only those landmarks with the highest potential pt (u). The restriction of the landmarks in use has the advantage that the calculation of the potential is faster while its quality is improved. An interesting observation is that using k landmarks is in fact very similar to using the maximum norm in a k-dimensional space. Each landmark corresponds to one dimension and, for a node, the distance to a landmark is the coordinate in the corresponding dimension. Such high-dimensional drawings have been used in [14], where they are projected to 2D using principal component analysis (PCA). This graph-drawing techniques has also been successfully used in [37] for goaldirected search and other geometric speed-up techniques. Distances from Graph Condensation For restricted shortest-path problems, performing a single run of an unrestricted Dijkstra’s algorithm is a relatively cheap operation. Examples are travel planning systems for scheduled vehicles like busses or trains. The complexity of the problem is much higher if you take connections, vehicle types, transfer times, or traffic days into account. It is therefore feasible to perform a shortest-path computation to find tighter lower bounds [29]. More precisely, you run Dijkstra’s algorithm on a condensed graph: The nodes of this graph are the stations (or stops) and an edge between two stations exists iff there is a non-stop connection. The edges are weighted by the minimal travel time. The distances of all v to the target t can be obtained by a single run of Dijkstra’s algorithm from the target t with reversed edges. These distances provide a feasible potential for the time-expanded graph, since the distances are a feasible potential in the condensed graph and an edge between two stations in the time-expanded graph is at least as long as the corresponding edge in the condensed graph. 3.3

Hierarchical Methods

This speed-up technique requires a preprocessing step at which the input graph G = (V, E) is enriched with additional edges representing shortest paths between

certain nodes. The additional edges can be seen as “bridges” or “short-cuts” for Dijkstra’s algorithm. These additional edges thereby realize new levels that step-by-step coarsen the graph. To find a shortest path between two nodes s and t using a hierarchy, it suffices for Dijkstra’s algorithm to consider a relatively small subgraph of the “hierarchical graph”. The hierarchical structure entails that a shortest path from s to t can be represented by a certain set of upward and of downward edges and a set of level edges passing at a maximal level that has to be taken into account. Mainly two methods have been developed to create such a hierarchy, the multi-level approach [33, 34, 18, 4] and highway hierarchies [31, 32]. These hierarchical methods are already close to the idea of using precomputed shortest paths tables for a small number of very frequently used “transit nodes”. Recently, this idea has been explored for the computation of shortest paths in road networks with respect to travel time [1]. Multi-Level Approach The decomposition of the graph can be realized using separators Si ⊂ V for each level, called selected nodes at level i: S0 := V ⊇ S1 ⊇ . . . ⊇ Sl . These node sets can be determined on diverse criteria. In a simple, but practical implementation, they consist of the desired numbers of nodes with highest degree in the graph. However, with domain-specific knowledge about the central nodes in the graph, better separators can be found. Alternatively, the planar separator theorem or betweenness centrality can be used to find small separators [18]. There are three different types of edges being added to the graph: upward edges, going from a node that is not selected at one level to a node selected at that level, downward edges, going from selected to non-selected nodes, and level edges, passing between selected nodes at one level. The weight of such an edge is assigned the length of a shortest path between the end-nodes. In [4] a further enhancement of the multi-level approach is presented, which uses a precomputed auxiliary graph with additional information. Instead of a single multi-level graph, a large number of small partial graphs is precomputed, which are optimized individually. This approach results in even smaller query times than achieved by the original multi-level approach. On the other hand, however, a comparably heavy preprocessing is required. Highway Hierarchies A different approach presented by [31, 32] is also based on the idea that only a “highway network” needs to be searched outside a the neighborhood of the source and the target node. Shortest path trees are used to determine a hierarchy. This has the advantage that no additional information like a separator is needed. Moreover, the use of highway hierarchies requires a less extensive preprocessing. The construction relies on a slight modification of Dijkstra’s algorithm that ensures that a sub-path ui , . . . , uj of a shortest path u1 , . . . , ui , . . . , uj , . . . , uk is always returned as the shortest path from ui to uj . These shortest paths are called canonical. Consider the sub-graph of G that consists of all edges in canonical shortest paths. The next level of the hierarchy is then induced by all nodes with degree at least two (i.e. the 2-core of the union of canonical shortest paths). Finally, nodes of degree 2 are then iteratively replaced by edges for a further contraction of the new level of the hierarchy.

3.4

Node and Edge Labels

Approaches based on node or edge labels use precomputed information as an indicator if a node or an edge has to be considered during an execution of Dijkstra’s algorithm for a certain target node t. Reach-Based Routing Reach-based routing prunes the search space based on a centrality measure called “reach” [13]. Intuitively, a node in the graph is important for shortest paths, if it is situated in the middle of long shortest paths. Nodes that are only at the beginning or the end of long shortest paths are less central. This leads to the following formal definition: Definition 2 (Reach). Given a weighted graph G = (V, E), l : E → R+ 0 and a shortest s-t path P , the reach on the path P of a node v ∈ P is defined as r(v, P ) := min{l(Psv ), l(Pvt )} where Psv and Pvt denote the sub-paths of P from s to v and from v to t, respectively. The reach r(v) of v ∈ V is defined as the maximum reach for all shortest s-t paths in G containing v. In a search for a shortest s-t path Pst , a node v ∈ V can be ignored, if (1) the distance l(Psv ) from s to v is larger than the reach of v and (2) the distance l(Pvt ) from v to t is larger than the reach of v. While performing Dijkstra’s algorithm, the first condition is easy to check, since l(Psv ) is already known. The second condition is fulfilled if the reach is smaller than a lower bound of the distance from v to t. (Suited lower bounds for the distance of a node to the target are already described for goal-directed search in Sect. 3.2.) Lines 7-13 of Algorithm 1 are therefore not performed if conditions (1) and (2) are surely fulfilled. To compute the reach for all nodes, we perform a single-source all-target shortest-path computation for every node. With a modified depth first search on the shortest-path trees, it is easy to compute the reach of all nodes using the following insight: For two shortest paths Psx and Psy with a common node v ∈ Psx and v ∈ Psy , we have max{r(v, Psx ), r(v, Psy )} = min{l(Psv ), max{l(Pvx ), l(Pvy )}}. The preprocessing for sparse graphs needs therefore O(n2 log n) time and O(n) space. In case such a heavy preprocessing is not acceptable, [13] also describes how to compute upper bounds for the reach. As mentioned in [11], the reach criterion can be extended to edges, which even improves its effectiveness but also increases the preprocessing time. Edge Labels This approach attaches a label to each edge that represents all nodes to which a shortest path starts with this particular edge [22, 23, 27, 33, 36, 38]. More precisely, we first determine, for each edge (u, v) ∈ E, the set S(u, v) of all nodes t ∈ V to which a shortest u-t path starts with the edge (u, v). The shortest path queries are then answered by Dijkstra’s algorithm restricted to those edges (u, v) for which the target node is in S(u, v). Similar to a traffic sign, the edge label shows the algorithm if the target node might

be in the target region of the edge. It is easy to verify that such a pruned shortest-path computation returns a shortest path: If (u, v) is part of a shortest s-t path, then its sub-path from u to t is also a shortest path. Therefore, t must be in S(u, v), because all nodes to which a shortest path starts with (u, v) are located in S(u, v). The restriction of the graph can be realized on-line during the shortest-path computation by excluding those edges whose edge label does not contain the target node (line 5 of algorithm 1). Geometric Containers As storing all sets S(u, v) would need O(n2 ) space, one can use a superset of S(u, v) that can be represented with constant size. Using constant-sized edge labels, the size of the preprocessed data is linear in the size of the graph. Given a layout L : V → R2 of the graph, an efficient and easy object type for an edge label associated to (u, v) is an enclosing geometric object of {L(t) | t ∈ S(u, v)}. Actually, the bounding box, i.e. the smallest rectangle parallel to the axes that contains {L(t) | t ∈ S(u, v)} turns out to be very effective as geometric container [38]. The bounding boxes can be computed beforehand by running a single-source all-target shortest-path computation for every node. The preprocessing for sparse graphs needs therefore O(n2 log n) time and O(n) space. Arc Flags If you drop the condition that the edge labels must have constant size, you can get much better however. An approach that performs very well in practice [22, 23, 27], is to partition the node set in p regions with a function r : V −→ {1, . . . , p}. Then an arc flag, i.e. a p-bit-vector where each bit represents one region is used as edge label. For an edge e, a region is marked in the pbit-vector of e if it contains a node v with v ∈ S(e).) Then the overall space requirement for the preprocessed data is Θ(p · m). But an advantage of bitvectors as edge labels is the insight that the preprocessing does not need to compute all -pairs shortest paths. Every shortest path from any node s outside a region R to a node inside a region R has to enter the region R at some point. As s is not a member of region R, there exists an edge e = (u, v) such that r(u) 6= r(v). It is therefore sufficient, if the preprocessing algorithm regards only the shortest paths to nodes v that are on the boundary of a region. These paths can be determined efficiently by a backward search starting at the boundary nodes. Usually, the number of boundary nodes is by orders of magnitude smaller than n. A crucial point for this type of edge labels is an appropriate partitioning of the node set. Using a layout of the graph, e.g. a grid, quad-trees or kd-trees can be used. In a general setup, a separator according to [21] is the best choice we are aware of [27]. 3.5

Combining Speed-Up Techniques

It has been shown in various publications [3, 11, 12, 16, 17, 30–33, 37] that the full power of speed-up techniques is unleashed, if various speed-up techniques are combined. In [16, 17] combinations of bidirectional search, goal-directed search, multi-level approach and geometric container are examined. For an experimental evaluation we refer to these papers. In this section, we concentrate on cases,

where an effective combination of two speed-up techniques is not obvious. The extension to a combination of three or four techniques is straight forward, once the problem of combining two of them is solved. However, not every combination is useful, as the search space may not be decreased (much) by adding a third or fourth speed-up techniques. Bidirectional Search and Goal-Directed Search Combining goal-directed and bidirectional search is not as obvious as it may seem at first glance. [30] provides a counter-example to show that simple application of a goal-directed search forward and a “source-directed” search backward yields a wrong termination condition. However, the alternative condition proposed there has been shown in [20] to be quite inefficient, as the search in each direction almost reaches the source of the other direction. An alternative is to use the same potential in both directions. With a potential from Sect. 3.2, you already get a speed-up (compared to using either goal-directed or bidirectional search). But one can do better using a combination of potentials: if ps (v) is a feasible potential for the backward search, then ps (t) − ps (v) is a feasible potential for the forward search (although not necessarily a good one). In order to balance the forward and the backward search, the average 12 (pt (v) + ps (t) − ps (v)) is a good compromise [10]. Bidirectional Search and Hierarchical Methods Basically, bidirectional search can be applied to the subgraph defined by the multi-level approach. In an actual implementation, that subgraph is computed on-the-fly during Dijkstra’s algorithm: for each node considered, the set of necessary outgoing edges is determined. If a bidirectional search is applied to the multi-level subgraph, a symmetric, backward version of the subgraph computation has to be implemented: for each node considered in the backward search, the incoming edges that are part of the subgraph have to be determined. See [16, 17] for an experimental evaluation. Actually, [31, 32] takes this combination even further in that it fully integrates the two approaches. The conditions for the pruning of the search space are interweaved with the fact that the search is performed in two directions at the same time. Bidirectional Search and Reach-Based Routing The reach criterion l(Psv ) ≤ r(v) ∨ l(Pvt ) ≤ r(v) can be used directly in the backward direction of the bidirectional search, too. In the backward search, l(Pvt ) is already known whereas we have to use a lower bound instead of l(Psv ) to replace the first condition l(Psv ) ≤ r(v). However, even without using a geometric lower bound but only the known distances for pruning, [11] reports good results. Bidirectional Search and Edge Labels In order to take advantage of edge labels in both directions of a bidirectional search, a second set of edge labels is needed. For each edge e ∈ E, we compute the set S(e) and the set Srev (e) of those nodes from which a shortest path ending with e exists. Then we store for each edge e ∈ E appropriate edge labels for S(e) and Srev (e). The forward search checks whether the target is contained S(e), the backward search, whether the source is in Srev (e). See [16, 17].

Goal-Directed Search and Highway Hierarchies Already the original highway algorithm [31, 32] accomplishes a bidirectional search. In [3] the highway hierarchies are further enhanced with goal-directed capabilities using potentials for forward and backward search based on landmarks. Unfortunately, the highway algorithm cannot abort the search as soon as an s-t path is found. However, another aspect of goal-directed search can be exploited, the pruning. As soon as an s-t path is found it yields an upper bound for the length of the shortest s-t path. Comparing upper and lower bound can then be used to prune the search. Altogether, the combination of highway hierarchies and landmarks brings less improvement than one might hope. On the other hand, using stopping the search as soon as an s-t path is found at the cost of losing correctness of the result (the s-t path found is not always the shortest s-t path) leads to an impressive speedup. Moreover, almost all paths found are also shortest and, in the rare other cases the approximation error is extremely small. Goal-Directed Search and Reach-Based Routing Goal-directed search can also be applied to the subgraph that is defined by the reach criterion. However, some care is needed if the subgraph is determined on-line (which is the common way to implement it) with the restriction by the reach. In particular, one should choose an implementation of goal-directed search that doesn’t change the distance labels of the nodes, as they are used to check the reach criterion. A detailed analysis of this combination can be found in [11]. Finally, in [12] the study of reachbased routing in combination with goal-directed search based on landmarks is continued.

4

Conclusion

We have summarized various techniques to speed-up Dijkstra’s algorithm. All of them guarantee to return a shortest path but run considerably faster. After all, the “best” choice of a speed-up technique heavily depends on the availability of a layout, the size of the main memory, the amount of preprocessing time you are willing to spend, and last but not least on the graph data considered.

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